I think I understand the question a little better now. When voltage increased, OP expected an exponential decay but instead saw a peaking to positive value and then going back down to negative value at which exponential decay happens.
Pretty much think of this as two current sources superimposing on each other. One is the current caused by difference in voltage (Ih). Ih is a mix of potassium and sodium current. While it is voltage dependent, it doesn't follow simple Ohm's law as this has to do with activation of channels rather than membrane. source The other is current caused by capacitive property of the membrane. Since capacitive current is proportional to rate of change in voltage, increase in voltage results in positive capacitive current.
So when voltage increased from -125 to -55 mV, there is a positive capacitive current as well as exponentially decaying Ih. These two superimposing each other almost leads to the picture we see in OP's question. I cannot explain why there is the huge negative spike in current after the small positive peak though.
Source: I learned this in class one year ago.
and nice animation of what happens during voltage clamp experiment